Resumo em Inglês:

The domain of validity of standard thermodynamics and Boltzmann-Gibbs statistical mechanics is discussed and then formally enlarged in order to hopefully cover a variety of anomalous systems. The generalization concerns nonextensive systems, where nonextensivity is understood in the thermodynamical sense. This generalization was first proposed in 1988 inspired by the probabilistic description of multifractal geometries, and has been intensively studied during this decade. In the present effort, after introducing some historical background, we briefly describe the formalism, and then exhibit the present status in what concerns theoretical, experimental and computational evidences and connections, as well as some perspectives for the future. In addition to these, here and there we point out various (possibly) relevant questions, whose answer would certainly clarify our current understanding of the foundations of statistical mechanics and its thermodynamical implications.

Resumo em Inglês:

We present some recent developments concerning general aspects of the thermodynamical formalism. Through simple arguments, we exhibit some basic entropies that have most of the thermodynamic properties of the Shannon entropy. Their stabilities are also analysed, and several points concerning nonextensive thermodynamics are discussed.

Resumo em Inglês:

The functional properties of the entropy gives rise to 6 possible types of thermodynamics. Additivity or superadditivity or subadditivity are closely related to extensivity and this is one of the characteristics used to distinguish the 6 types. We give examples (some are somewhat academic) of all of these, except one. For this purpose we draw also on black hole systems which have been proposed. Some of these systems have subadditive entropies, i.e. they tend to fragment rather than clump. After proposing a new entropy function we raise the problem of how to select from these entropy functions. Are some better than others?

Resumo em Inglês:

The role of Tsalli's non-extensive Information Measure within an à la Jaynes Information-Theory-based formulation of Statistical Mechanics is discussed in rather detailed fashion.

Resumo em Inglês:

We develop four identities concerning parameter differentiation of fractional powers of operators appearing in the Tsallis ensembles of quantum statistical mechanics of nonextensive systems. In the appropriate limit these reduce to the corresponding differentiation identities of exponential operators of the Gibbs ensembles of extensive systems derived by Wilcox.

Resumo em Inglês:

Traditional field-theoretical methods to study extensive many-particle systems are generalized to discuss nonextensive situations. In particular, generalizations of Green functions, path integral, and Gaussian integration are performed in the context of nonextensive Tsallis statistical mechanics. These developments employ integral representations that connect the usual and the generalized cases.

Resumo em Inglês:

We revisit Tsallis Maximum Entropy Solutions to the Vlasov-Poisson Equation describing gravitational N-body systems. We review their main characteristics and discuss their relationship with other applications of Tsallis statistics to systems with long range interactions. In the following considerations we shall be dealing witha D-dimensional space so as to be in a position to investigate possible dimensional dependences of Tsallis' parameter q. The particular and important case of the Schuster solution is studied in detail, and the pertinent Tsallis parameter q is given as a function of the space dimension. In the special case of three dimensional space we recover the value q = 7/9, that has already appeared in many applications of Tsallis' formalism involving long range forces.

Resumo em Inglês:

Tsallis has proposed a generalization of Boltzmann-Gibbs thermostatistics by introducing a family of generalized nonextensive entropy functionals with a single parameter q. These reduce to the extensive Boltzmann-Gibbs form for q = 1, but a remarkable number of statistical and thermodynamic properties have been shown to be q-invariant - that is, valid for any q. In this paper, we address the question of whether or not the value of q for a given viscous, incompressible fluid can be ascertained solely by measurement of the fluid's hydrodynamic properties. We find that the hydrodynamic equations expressing conservation of mass and momentum are q-invariant, but the conservation of energy is not. Moreover, we find that ratios of transport coefficients may also be q-dependent. These dependences may therefore be exploited to measure q experimentally.

Resumo em Inglês:

It is shown that Tsallis's generalized statistics provides a natural frame for the statistical-thermodynamical description of anomalous diffusion. Within this generalized theory, a maximum-entropy formalism makes it possible to derive a mathematical formulation for the mechanisms that underly Lévy-like superdiffusion, and for solving the nonlinear Fokker-Planck equation.

Resumo em Inglês:

A particular class of strong non-Markovian stochastic processes have been studied by using a characteristic functional technique previously reported. Exact results for all moments and the whole Kolmogorov hierarchy are presented. The asymptotic scaling of the non-Markovian stochastic process has been characterized in terms of the long-range correlated noise appearing in the correponding stochastic differential equation. A generalized Wiener process has therefore been completely characterized, its power spectrum and fractal dimensions have been studied and its possible connection with the q-statistics has been pointed out.

Resumo em Inglês:

Within the two state theory (TST) for stochastic resonance (SR) we analize two different aspects: (a) the extension of the TST in order to include potential asymmetry (i.e.: the states show different stabilities); (b) the evaluation of transition rates for systems whose stationary distribution is non Gaussian. We apply the results of (a) to study the role of the potential symmetry for SR in bistable systems, observing that the signal-to-noise ratio increases with the symmetry of the potential of the system indicating that it is this feature that governs the optimization of the response. We apply the results of (b) to discuss SR in situations where we can assume that the noise is non Gaussian, and discuss its relation with experimental results in sensory systems.

Resumo em Inglês:

Low dimensional non-linear maps are prototype models to study the emergence of complex behavior in nature. They may exhibit power-law sensitivity to initial conditions at the edge of chaos which can be naturally formulated within the generalized Tsallis statistics prescription which is characterized by the entropic index q. General scaling arguments provide a direct relation between the entropic index q and the scaling exponents associated with the extremal sets of the multifractal critical attractor. The above result comes in favor of recent conjectures that Tsallis statistics is the natural frame for studying systems with fractal-like structure in the phase-space. Power law sensitivity in high-dimensional dissipative and Hamiltonian systems are also discussed within the present picture.

Resumo em Inglês:

The physics of nuclear reactions in stellar plasma is reviewed with special emphasis on the importance of the velocity distribution of ions. Then the properties (density and temperature) of the weak-coupled solar plasma are analysed, showing that the ion velocities should deviate from the Maxwellian distribution and could be better described by a weakly-nonextensive (|q-1| < 0.02) Tsallis' distribution. We discuss concrete physical frameworks for calculating this deviation: the introduction of higher-order corrections to the diffusion and friction coefficients in the Fokker-Planck equation, the influence of the electric-microfield stochastic distribution on the particle dynamics, a velocity correlation function with long-time memory arising from the coupling of the collective and individual degrees of freedom. Finally, we study the effects of such deviations on stellar nuclear rates, on the solar neutrino fluxes, and on the pp neutrino energy spectrum, and analyse the consequences for the solar neutrino problem.

Resumo em Inglês:

Consequences of long-range hopping in one-dimensional tight-binding models are studied. a hopping term proportional to <img src="http:/img/fbpe/bjp/v29n1/liab1.gif" alt="liab1.gif (146 bytes)" align="absmiddle"> is used, where rij denotes the distance between atoms i and j and <FONT FACE="Symbol">a</font> determines the range of the interactions within the system. Calculations of the diffusion of an electron along the lattice yield interesting effects of nonextensivity. In particular, we find that the mean square displacement scales anomalously as Dt<FONT FACE="Symbol">g</font> in the following way: For 0 < <FONT FACE="Symbol">a</font> < 1, we find D <FONT FACE="Symbol">µ</font> NN*, where N is the number of atoms on the lattice and <img src="http:/img/fbpe/bjp/v29n1/liab2.gif" alt="liab2.gif (238 bytes)" align="absmiddle"> is related to the number of elements interacting at a given <FONT FACE="Symbol">a</font>. In this regime the behaviour is subdiffusive (.5 <FONT FACE="Symbol">£ g</font> < 1) but approaches normal diffusion (<FONT FACE="Symbol">g</font> = 1) for <FONT FACE="Symbol">a</font> = 1. There exists a transition region between 1 < <FONT FACE="Symbol">a</font> < 2, where the diffusion coefficient loses its system size dependency and becomes size independent for all <FONT FACE="Symbol">a</font> <FONT FACE="Symbol">³ </font>2. In addition, we find 1< <FONT FACE="Symbol">g</font> <FONT FACE="Symbol">£ </font>2 (superdiffusion) for <FONT FACE="Symbol">a</font> >1. Ballistic motion (<FONT FACE="Symbol">g</font> = 2) is recovered for all <FONT FACE="Symbol">a</font> <FONT FACE="Symbol">³ </font>1.5 and is maintained in the nearest neighbour limit. Specific heat and internal energy as a function of temperature and system size are also analyzed. They appear extensive on the macroscopic level for all values of <FONT FACE="Symbol">a</font>.

Resumo em Inglês:

Tsallis's generalization of statistical mechanics is summarized. A modification of this formalism which employs a normalized expression of the q-expectation value for the computation of equilibrium averages is reviewed for the cases of pure Tsallis statistics and Maxwell-Tsallis statistics. Monte Carlo and Molecular Dynamics algorithms which sample the Tsallis statistical distributions are presented. These methods have been found to be effective in the computation of equilibrium averages and isolation of low lying energy minima for low temperature atomic clusters, spin systems, and biomolecules. A phase space coordinate transformation is proposed which connects the standard Cartesian positions and momenta with a set of positions and momenta which depend on the potential energy. It is shown that pure Tsallis statistical averages in this transformed phase space result in the q-expectation averages of Maxwell-Tsallis statistics. Finally, an alternative novel derivation of the Tsallis statistical distribution is presented. The derivation begins with the classical density matrix, rather than the Gibbs entropy formula, but arrives at the standard distributions of Tsallis statistics. The result suggests a new formulation of imaginary time path integrals wich may lead to an improvement in the simulation of equilibrium quantum statistical averages.

Resumo em Inglês:

We review uses of Tsallis statistical mechanics in the protein folding problem. Monte Carlo simulated annealing algorithm and generalized-ensemble algorithm with both Monte Carlo and stochastic dynamics algorithms are discussed. Simulations by these algorithms are performed for a penta peptide, Met-enkephalin. In particular, for generalized-ensemble algorithms, it is shown that from only one simulation run one can find the global-minimum-energy conformation and obtain probability distributions in canonical ensemble for a wide temperature range, which allows the calculation of any thermodynamic quantity as a function of temperature.

Resumo em Inglês:

A hybrid approach is described, which combines stochastic classical molecular dynamics and first principles Density Functional theory to model the atomic and electronic structure of large molecular and solid-state systems. The stochastic molecular dynamics using Generalized Simulated Annealing (GSA) is based on the nonextensive statistical mechanics and thermodynamics. Examples are given of applications in linear-chain polymers, structural ceramics, impurities in metals, and pharmacological molecule-protein interactions.